Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Media Studies: "Media, History and Society", and "Thinking Popular Culture".

“When cultural life is re-defined as a perpetual round of entertainments, when serious public conversation becomes a form of baby talk, when, in short, a people become an audience and their public business a vaudeville act, then a nation finds itself at risk.” (Postman) The dire tones of Postman quoted in Janet Cramer’s Media, History, Society: A Cultural History of US Media introduce one view that she canvasses, in the debate of the moment, as to where popular culture is heading in the digital age. This is canvassed, less systematically, in Thinking Popular Culture: War Terrorism and Writing by Tara Brabazon, who for example refers to concerns about a “crisis of critical language” that is bothering professionals—journalists and academics or elsewhere—and deplores the advent of the Internet, as a “flattening of expertise in digital environments”.